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  2. Completing the square - Wikipedia

    en.wikipedia.org/wiki/Completing_the_square

    One way to see this is to note that the graph of the function f(x) = x 2 is a parabola whose vertex is at the origin (0, 0). Therefore, the graph of the function f(x − h) = (x − h) 2 is a parabola shifted to the right by h whose vertex is at (h, 0), as shown in the top figure.

  3. Quadratic function - Wikipedia

    en.wikipedia.org/wiki/Quadratic_function

    To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots r 1 and r 2. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.

  4. Quadratic formula - Wikipedia

    en.wikipedia.org/wiki/Quadratic_formula

    The roots of the quadratic function y = ⁠ 1 / 2 ⁠ x 2 − 3x + ⁠ 5 / 2 ⁠ are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.

  5. Basic feasible solution - Wikipedia

    en.wikipedia.org/wiki/Basic_feasible_solution

    Here, m=2 and there are 10 subsets of 2 indices, however, not all of them are bases: the set {3,5} is not a basis since columns 3 and 5 are linearly dependent. The set B={2,4} is a basis, since the matrix = is non-singular.

  6. Quadratic form - Wikipedia

    en.wikipedia.org/wiki/Quadratic_form

    An integral quadratic form has integer coefficients, such as x 2 + xy + y 2; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning Q(x, y) ∈ Z if x, y ∈ Λ.

  7. Parabola - Wikipedia

    en.wikipedia.org/wiki/Parabola

    In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2y 2. Generalizations to more variables yield ...

  8. Polynomial - Wikipedia

    en.wikipedia.org/wiki/Polynomial

    For example, the term 2x in x 2 + 2x + 1 is a linear term in a quadratic polynomial. The polynomial 0, ... For example, antiderivatives of x 2 + 1 have the form ...

  9. Cubic function - Wikipedia

    en.wikipedia.org/wiki/Cubic_function

    The points P 1, P 2, and P 3 (in blue) are collinear and belong to the graph of x 3 + ⁠ 3 / 2 ⁠ x 2 − ⁠ 5 / 2 ⁠ x + ⁠ 5 / 4 ⁠. The points T 1, T 2, and T 3 (in red) are the intersections of the (dotted) tangent lines to the graph at these points with the graph itself. They are collinear too.